Full Length Paper Series A

Mathematical Programming

, Volume 144, Issue 1, pp 1-38

First online:

Iteration complexity of randomized block-coordinate descent methods for minimizing a composite function

  • Peter RichtárikAffiliated withSchool of Mathematics, University of Edinburgh Email author 
  • , Martin TakáčAffiliated withSchool of Mathematics, University of Edinburgh

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In this paper we develop a randomized block-coordinate descent method for minimizing the sum of a smooth and a simple nonsmooth block-separable convex function and prove that it obtains an \(\varepsilon \)-accurate solution with probability at least \(1-\rho \) in at most \(O((n/\varepsilon ) \log (1/\rho ))\) iterations, where \(n\) is the number of blocks. This extends recent results of Nesterov (SIAM J Optim 22(2): 341–362, 2012), which cover the smooth case, to composite minimization, while at the same time improving the complexity by the factor of 4 and removing \(\varepsilon \) from the logarithmic term. More importantly, in contrast with the aforementioned work in which the author achieves the results by applying the method to a regularized version of the objective function with an unknown scaling factor, we show that this is not necessary, thus achieving first true iteration complexity bounds. For strongly convex functions the method converges linearly. In the smooth case we also allow for arbitrary probability vectors and non-Euclidean norms. Finally, we demonstrate numerically that the algorithm is able to solve huge-scale \(\ell _1\)-regularized least squares problems with a billion variables.


Block coordinate descent Huge-scale optimization Composite minimization Iteration complexity Convex optimization LASSO Sparse regression Gradient descent Coordinate relaxation Gauss–Seidel method

Mathematics Subject Classification (2000)

65K05 90C05 90C06 90C25